Also, before you install a new custom ROM, you can make full backup of your apps through Titanium Backup, use GMail to backup your contacts (make a dummy GMail account if you don’t have one), and use any SMS Text backup app from the Play Store to backup your SMS texts.

Here is the Step By Step Instruction about how to Install a custom ROM on Rooted Samsung Galaxy SIII i9300 device

Step 1. First, go ahead and download the ROM zip file and copy the whole file over to your phone’s internal storage or microSD card. DO NOT UNZIP the ROM zip file. Mac OSX usually automatically unzips the zip file but just use the zip file not the folder.

Sometimes there may be several zip files, usually the ROM and Gapps or other mods. Copy all of them over to your phone’s internal storage or microSD card. For multiple files, you simply have to install the ROM first then the rest.

Step 2. First, reboot into ClockworkMod Recovery by holding down Volume Up, Center Home, and Power buttons together for 10 seconds.

Step 3. Choose “wipe data/factory reset”.

Step 4. Choose “Yes”.

Step 5. Choose “install zip from sdcard”.

Step 6. Choose “choose zip from sdcard” OR “choose zip from internal sdcard” depending on where you copied the ROM zip file to in Step 1.

Step 7. Browse and find the ROM zip file you copied over to your phone in Step 1. In the example below, I am going to choose “I9300_Omega_v4.2_XXALEF.zip” which is the ROM zip file for Omega ROM.

Step 8. Choose “Yes”.

Step 9. Depending on the ROM you install there may be an additional graphical installer. If it does, simply follow the instructions on the screen, otherwise just give it 5-10 minutes for the ROM to install.

Step 10. Reboot and you should find your phone with a new custom ROM!

Step 11. (Optional) If for some reason your phone doesn’t boot into your new custom ROM or you get a lot of force closes or you can’t download apps from the Play Store, you can clear cache and dalvik cache in CWM.
Cache and Dalvik cache are temporary files in your phone that can go corrupt while installing a new ROM. It doesn’t hurt to format/wipe cache so let’s do that.
Reboot into ClockworkMod Recovery by holding down Volume Up, Center Home, and Power buttons together for 10 seconds.

Step 12. Choose “Wipe cache”.

Step 13. Choose “Yes”.

Step 14. Also you can wipe dalvik cache, which is usually in Advanced menu.

Step 15. 99% of the time, wiping cache and dalvik cache will solve your bootloop problems and force closes. You don’t have to do this every time you install a new ROM (some people do) but just know this is the FIRST thing you do when something goes wrong while installing a new ROM.